1

Optimal Multicast Capacity and Delay Tradeoffsin MANETs

Jinbei Zhang, Xinbing Wang, Senior Member, IEEE , Xiaohua Tian, Member, IEEEYun Wang, Xiaoyu Chu, and Yu Cheng, Senior Member, IEEE

Abstract—In this paper, we give a global perspective of multicast capacity and delay analysis in Mobile Ad Hoc Networks (MANETs).Specifically, we consider four node mobility models: (1) two-dimensional i.i.d. mobility, (2) two-dimensional hybrid random walk, (3)one-dimensional i.i.d. mobility, and (4) one-dimensional hybrid random walk. Two mobility time-scales are investigated in this paper: (i)Fast mobility where node mobility is at the same time-scale as data transmissions; (ii) Slow mobility where node mobility is assumedto occur at a much slower time-scale than data transmissions. Given a delay constraint D, we first characterize the optimal multicastcapacity for each of the eight types of mobility models, and then we develop a scheme that can achieve a capacity-delay tradeoff closeto the upper bound up to a logarithmic factor. In addition, we also study heterogeneous networks with infrastructure support.

Index Terms—Multicast capacity and delay tradeoffs, Mobile Ad Hoc Networks (MANETs), independent and identically distributed(i.i.d.) mobility models, hybrid random walk mobility models, capacity achieving schemes, heterogeneous networks

F

1 INTRODUCTION

Since the seminal paper by Gupta and Kumar [1], wherea maximum per-node throughput of O(1/

√n) was es-

tablished in a static network with n nodes, there hasbeen tremendous interest in the networking researchcommunity to understand the fundamental achievablecapacity in wireless ad hoc networks. How to improvethe network performance, in terms of the capacity anddelay, has been a central issue.

Many works have been conducted to investigate theimprovement by introducing different kinds of mobilityinto the network, [2], [3], [4], [5], [6], [7]. Other worksattempt to improve capacity by introducing base stationsas infrastructure support, [8], [9], [10].

As the demand of information sharing increases rapid-ly, multicast flows are expected to be predominant inmany of the emerging applications, such as the orderdelivery in battlefield networks and wireless video con-ferences. Related works are [11], [12], [13], [14], [15], [16],[17], [18], [19], [20], [21], [22], including static, mobileand hybrid networks.

Introducing mobility into the multicast traffic pattern,Hu et al. [14] studied a motioncast model. Fast mo-bility was assumed. Capacity and delay were calculat-ed under two particular algorithms, and the tradeoffderived from them was λ = O( D

nk log k ), where k was

• Jinbei Zhang, Xinbing Wang, Yun Wang, and Xiaoyu Chu are with theDepartment of Electronic Engineering, Shanghai Jiao Tong University,China. Jinbei Zhang is also with the State Key Laboratory of IntegratedServices Networks, Xidian University, Xi’an, China. Xinbing Wang is alsowith National Mobile Communications Research Laboratory, Southeast U-niversity, Nanjing, China. E-mail: abelchina, xwang8, xtian, sunshinehk,cxygrace@sjtu.edu.cn.

• Yu Cheng is with the Department of Electrical and Computer Engineering,Illinois Institute of Technology, USA. E-mail: cheng@iit.edu.Correspondence author: Dr. Xinbing Wang.

the number of destinations per source. In their work,the network is partitioned into Θ(n) cells similar to[3] and TDMA scheme is used to avoid interference.Zhou and Ying [15] also studied the fast mobility modeland provided an optimal tradeoff under their networkassumptions. Specifically, they considered a networkthat consists of ns multicast sessions, each of whichhad one source and p destinations. They showed thatgiven delay constraint D, the capacity per multicastsession was O

min

n1, (log p)(log(nsp))

ÈDns

o. Then a

joint coding/scheduling algorithm was proposed to achievea throughput of O

min

n1,È

Dns

o. In their network,

each multicast session had no intersection with othersand the total number of mobile nodes was n = ns(p+1).

Heterogeneous networks with multicast traffic patternwere studied by Li et al. [16] and Mao et al. [17]. Wiredbase stations are used and their transmission range cancover the whole network. Li et al. [18] studied a densenetwork with fixed unit area. The helping nodes in theirwork are wireless, but have higher power and only actas relays instead of sources or destinations. [16], [17] and[18] all study static networks.

In this paper, we give a general analysis on the optimalmulticast capacity-delay tradeoffs in both homogeneousand heterogeneous MANETs. We assume a mobile wire-less network that consists of n nodes, among which ns =ns nodes are selected as sources and nd = nα destinednodes are chosen for each. Thus, ns multicast sessionsare formed. Our results in homogeneous network arefurther used to study the heterogeneous network, wherem = nβ base stations connected with wires are uniformlydistributed in the unit square.

The purpose of this paper is to conduct extensive anal-ysis on the multicast capacity-delay tradeoff in mobilewireless networks. We study a variety of mobility models

iEEE TRANSACTIONS ON MOBILE COMPUTING VOL:PP NO:99

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which are also widely adopted in previous works. Theresults obtained may provide valuable insights on howmulticast will affect the network performance comparedto unicast networks, e.g., [4] [5]. By removing some limi-tations and constraints, we try to present a fundamentaland more general result than previous works.

We summarize our main results as follows where thelogarithmic factors are omitted here:

1) Two-dimensional i.i.d. mobility models: Under adelay constraint D, the maximum throughput permulticast session is O

n

nsnd

ÈDn nd

for fast mobil-

ity, and O

nnsnd

3

ÈDn nd

for slow mobility.

2) Two-dimensional hybrid random walk mobilitymodels: When B = o(1) and D = ω(| logB|/B2),the maximum throughput per multicast ses-sion is O

n

nsnd

ÈDn nd

for fast mobility, and

O

nnsnd

3

ÈDn nd

for slow mobility.

3) One-dimensional i.i.d. mobility models: Under adelay constraint D, the maximum throughput permulticast session is O

n

nsnd

3

ÈD2

n n2d

for fast mo-

bility, and O

nnsnd

4

ÈD2

n n2d

for slow mobility.

4) One-dimensional hybrid random walk mobilitymodels: When B = o(1) and D = ω(1/B2),the maximum throughput per multicast ses-sion is O

n

nsnd

3

ÈD2

n n2d

for fast mobility, and

O

nnsnd

4

ÈD2

n n2d

for slow mobility.

5) Heterogeneous networks with infrastructure sup-port:

a) In infrastructure mode, the maximum aggre-gate input throughput of the whole networkis Ti = minTi1, Ti2, where Ti1 is the capacityfrom sources to base stations while Ti2 is thedownlink capacity.

b) The aggregate input capacity of the hetero-geneous networks under the above eight d-ifferent kinds of mobility models is T =

max¦Ti, nsλa

©, where λa is the per session

capacity in each of the homogeneous networkspresented above.

The rest of the paper is organized as follows. In Section2, we outline the system models. The eight mobilitymodels in homogeneous networks are investigated inSection 3 to Section 8 respectively. Section 9 offers somediscussions on the obtained results. In Section 10 westudy the capacity of heterogeneous networks. In theend, we conclude this paper.

2 SYSTEM MODELS

We consider a mobile ad hoc network where n nodesmove within a unit square. Among them, ns nodes areselected as sources, and each node has nd distinct des-tinations. We group each source and its nd destinationsas a multicast session. Note that a particular node may

serves as both a source and a destination in differentmulticast sessions. Protocol Model [1] is employed. Thedefinitions of capacity and delay are also similar toprevious works, such as [1], [4] and [5].

2.1 Homogeneous NetworksMobile ad hoc network model: Consider an ad hocnetwork where n wireless mobile nodes are randomlydistributed in a unit square. The unit square is assumedto be a torus to avoid the border effect. We will study thefollowing mobility models, similar to [5], in this paper.

1) Two-dimensional i.i.d. mobility model:a) At the beginning of each time slot, nodes will

be uniformly and randomly distributed in theunit square.

b) The node positions are independent of eachother, and independent from time slot to timeslot.

2) Two-dimensional hybrid random walk model:Consider a unit square which is further dividedinto 1/B2 squares of equal size. Each of the smallersquare is called a RW-cell (random walk cell), andindexed by (Ux, Uy) where Ux, Uy ∈ 1, . . . , 1/B.A node which is in one RW-cell at a time slotmoves to one of its eight adjacent RW-cells orstays in the same RW-cell in the next time-slotwith a same probability. Two RW-cells are saidto be adjacent if they share a common point. Thenode position within the RW-cell is randomly anduniformly selected.

3) One-dimensional i.i.d. mobility model:a) Reasonably, we assume the number of mobile

nodes n and source nodes ns are both evennumbers. Among the mobile nodes, n/2 n-odes (including ns/2 source nodes), named H-nodes, move horizontally; and the other n/2nodes (including the other ns/2 source nodes),named V-nodes, move vertically.

b) Let (xi, yi) denote the position of node i. Ifnode i is a H-node, yi is fixed and xi is ran-domly and uniformly chosen from [0, 1]. Wealso assume that H-nodes are evenly distribut-ed vertically, so yi takes values 2/n, 4/n, . . . , 1.V-nodes have similar properties.

c) Assume that source and destinations in thesame multicast session are the same type ofnodes. Also assume that node i is a H-node ifi is odd, and a V-node if i is even.

d) The orbit distance of two H(V)-nodes is de-fined to be the vertical (horizontal) distanceof the two nodes.

4) One-dimensional hybrid random walk model:Each orbit is divided into 1/B RW-intervals (ran-dom walk interval). At each time slot, a nodemoves into one of two adjacent RW-intervals orstays at the current RW-interval. The node positionin the RW-interval is randomly, uniformly selected.

3

We further assume that at each time slot, at most Wbits can be transmitted in a successful transmission.

Mobility time scales: Two time scales of mobility areconsidered in this paper:

• Fast mobility: The mobility of nodes is at the sametime scale as the transmission of packets, i.e., in eachtime-slot, only one transmission is allowed.

• Slow mobility: The mobility of nodes is much s-lower than the transmission of packets, i.e., multipletransmissions may happen within one time-slot.

Scheduling Policies: We assume that there exists ascheduler that has all the information about the currentand past status of the network, and can schedule anyradio transmission in the current and future time slots,similar to [4]. We say a packet p is successfully deliveredif and only if all destinations within the multicast sessionhave received the packet. In each time slot, for eachpacket p that has not been successfully delivered andeach of its unreached destination k, the scheduler needsto perform the following two functions:

• Capture: The scheduler needs to decide whether todeliver packet p to destination k in the current timeslot. If yes, the scheduler then needs to choose onerelay node (possibly the source node itself) that hasa copy of the packet p at the beginning of the time-slot, and schedules radio transmissions to forwardthis packet to destination k within the same time-slot, using possibly multi-hop transmissions. Whenthis happens successfully, we say that the chosenrelay node has successfully captured the destinationk of packet p. We call this chosen relay node the lastmobile relay for packet p and destination k. And wecall the distance between the last mobile relay andthe destination as the capture range.

• Duplication: For a packet p that has not been suc-cessfully delivered, the scheduler needs to decidewhether to duplicate packet p to other nodes that donot have the packet at the beginning of the time-slot.The scheduler also needs to decide which nodes torelay from and relay to, and how.

2.2 Heterogeneous NetworksWe introduce m regularly placed base stations (con-nected with each other via wires) into the mobile adhoc networks and generate a heterogeneous network.Specifically, the base stations are placed at positions( 12√m+i 1√

m, 12√m+j 1√

m) with 0 ≤ i, j ≤

√m−1. Clearly,

these m regularly distributed base stations divide theoriginal square region into m subregions with side length1√m

. Here we assume that m is the square of some integerfor simplicity.

All transmissions can be carried out either in ad hocmode or in infrastructure mode (see Figure 1). We assumethat the base stations have a same transmission band-width, denoted by Wi for each. The bandwidth for eachmobile ad hoc node is denoted by Wa. Further, we evenlydivide the bandwidth Wi into two parts, one for uplink

transmissions and the other for downlink transmissions,so that these different kinds of transmissions will notinterfere with each other.

A transmission in infrastructure mode is carried outin the following steps:

1) Uplink: A mobile node holding packet p is select-ed, and transmits this packet to the nearest basestation.

2) Infrastructure relay: Once a base station receives apacket from a mobile node, all the other m − 1base stations share this packet immediately, (i.e.,the delay is considered to be zero) since all basestations are connected by wires.

3) Downlink: Each base station searches for all thepackets needed in its own subregion, and transmitall of them to their destined mobile nodes. At thisstep, every base station will adopt TDMA schemesto delivere different packets for different multicastsessions.

1

1/√m

BS

UplinkDownlink

Ad hoc

Fig. 1. Heterogeneous network with infrastructure sup-port.

3 TWO DIMENSIONAL I.I.D. FAST MOBILITYMODEL

In this section, we present the upper bound on multicastcapacity-delay tradeoff under the two-dimensional i.i.d.fast mobility model, and then propose a scheme to achievea capacity close to the upper bound up to a logarithmicfactor.

3.1 Upper BoundConsider packet p and one of its destinations k. Let Lp,k

denote the capture range for packet p and destinationk, Lp denote the capture range for packet p and itslast reached destination. Let Dp,k denote the number oftime slots it takes to reach destination k after reachingdestination k − 1. Denote Dp as the number of timeslots it takes to reach the last destination of packetp, which means Dp =

Pk Dp,k. And let Rp,k and Rp

denote the number of mobile relays holding packet p

4

when the packet reaches its k-th destination and lastdestination respectively. To reach a new destination k,all the nodes holding packet p should move across afraction of network area in Dp,k timeslots. Then we havethe following lemma.

Lemma 1: Under two-dimensional i.i.d. mobility modeland concerning successful encounter, the following in-equality holds for any causal scheduling policy,

c1 log nE[Dp,k] ≥1

(nd − k + 1)E[Lp,k] +

1n2

2E[Rp,k]

,

(1)where c1 is a positive constant.

Consider a large time interval T . The total number ofpackets communicated among all sessions is λnsT . Thenwe have the following lemma.

Lemma 2: Under fast mobility model and concerningnetwork radio resource consumption, the following in-equality holds for any causal scheduling policy (c2 is apositive constant),

λnsTXp=1

∆2

4

E[Rp]− nd

n+

λnsTXp=1

ndXk=1

π∆2

4E[L2

p,k] ≤ c2WT log n.

(2)

Proof: Here are some intuitive explanations. Proofsare similar to and can be easily inferred from AppendixB in [4].

We try to measure how much radio resource eachtransmission consumes, by calculating the areas of dis-joint disks caused by interference. Radio resource con-sumption is divided into two parts, Capture and Dupli-cation.

• Capture: For each packet p and each of its desti-nation k, the one-hop capture1 consumes an area ofπ∆2

4 (Lp,k)2. Hence, the upper bound on the expected

area consumed by all nd successful captures ofpacket p is

Pnd

k=1π∆2

4 E[L2p,k].

• Duplication: If the radius of transmission range is s,then w.h.p. there are πs2n nodes which can receivethe broadcast packets, and a disk of area π∆2

4 s2

centered at the transmitter will be disjoint fromothers. Therefore, we can use ∆2

4E[Rp]−nd

n as anupper bound on the expected area consumed byproducing Rp − nd copies of the packet to othernodes before any of them or the source itself suc-cessfully forwards the packet to the last destination.Note that since we use cooperative mode [14], wheredestinations can also act as relays, the copies pro-duced in Duplication should not only exclude thesource node but also exclude the nd−1 destinationswhich receive the copies in Capture procedure.

Theorem 1: Under two-dimensional i.i.d. fast mobilitymodel, let D be the mean delay averaged over all packets,

1. Concerning the multi-hop capture, consumption area is summed upby each hop transmission.

and let λ be the capacity per multicast session. Thefollowing upper bound holds for any causal schedulingpolicy,

λ ≤ min

(Θ(1),Θ

n

nsnd

rndD

nlog3 n

). (3)

Proof: Each source can send out at most W size ofpacket per time-slot, i.e., λ ≤ W = Θ(1). Therefore, weonly need to prove the second part.

From Lemma 1, we have

ndXk=1

(E[Lp,k] +1

n2)2 ≥

ndXk=1

1

(nd − k + 1)E[Rp,k]E[Dp,k] log n

≥ 1

E[Rp] log n

ndXk=1

1

(nd − k + 1)E[Dp,k]

≥ 1

E[Rp] log n

Pnd

k=11√

nd−k+1

2Pnd

k=1 E[Dp,k]

≥ 1

E[Rp] log n· c

′

2nd

E[Dp],

(4)where c

′

2 is a constant. Note thatPnd

k=11√

nd−k+1

2=

Θ(nd) when nd = Ω(1). Equations will hold when Rp,k =Θ(Rp) and

√nd − k + 1Dp,k = Θ(

√nd − i+ 1Dp,i) for all

k and i.There are two cases we need to consider.

Case 1: WhenPnd

k=1 E[L2p,k] = Ω(nd

n4 ), from Lemma 2,

λnsTXp=1

∆2

4

E[Rp]

n+

λnsTXp=1

ndXk=1

π∆2

4E[L2

p,k]

≥λnsTXp=1

∆2

4

E[Rp]

n+ c

′

3

λnsTXp=1

1

E[Rp] log n· nd

E[Dp]

≥ c′

1

λnsTXp=1

rnd

nE[Dp] log n

= c′

1

Énd

n log n

λnsTXp=1

Ê1

E[Dp]

≥ c′

1

Énd

n log n

(PλnsT

p=1 1)2PλnsTp=1

ÈE[Dp]

≥ c′

1

Énd

n log n

(PλnsT

p=1 1)2qPλnsTp=1 E[Dp] ·

PλnsTp=1 1

= c′

1

Énd

n log n

λnsT√D

(5)

where c′

1 and c′

3 are both constants. Equations hold whenRp = Θ(

Ènnd

Dp logn ) and Dp = Θ(Di) . Therefore,

Énd

n log n

λnsT√D

− λnsTnd

n≤ WT log n. (6)

5

When D = O( nnd logn ), the first term dominates and

hence,

λ ≤

ÈnD log3 n

ns√nd

. (7)

Case 2: WhenPnd

k=1 E[L2p,k] = O(nd

n4 ), the first term inthe left part of Lemma 2 would dominate. We have

4c2WT log n

∆2≥ 1

4c1 log n

λTnsn3

D− λT

nsnd

n.

Hence, for n large enough,

λ ≤ 16c1c2W

∆2

D log2 n

nsn3. (8)

Finally, we compare the two inequalities we have ob-tained, i.e., (4) and (8). When D = O( n

nd logn ), inequality(8) will eventually be the loosest for large n, the optimalcapacity-delay tradeoff is upper bounded by

λ ≤ Θ

n

nsnd

ÊD log3 n

nnd

!.

3.2 Achievable Lower BoundIn this subsection, we will show how the study of theupper bound also helps us in developing a new schemethat can achieve a capacity-delay tradeoff that is close tothe upper bound.

Choosing Optimal Values of Key Parameters:From Theorem 1, we have

λ = O

n

nsnd

ÊndD log3 n

n

= O(n− 2s+α−1−d

2 log32 n).

In order to achieve the maximum capacity on the righthand side, all inequalities in the proof of Theorem 1should hold with equality. By studying the conditionsunder which these inequalities are tight, we are ableto identify that the optimal choices of various key pa-rameters of the scheduling policy. We can infer that theparameters (such as E[Rp,k], E[Lp,k]) of each packet p andeach destination node k should be the same and concen-trate on their respective average values. This implies thatthe scheduling policy should use the same parametersfor all packets and all destinations. We further assumethat ns = ns, 0 ≤ s ≤ 1; nd = nα, 0 ≤ α ≤ 1 andD = nd, 0 ≤ d < 1 − α. In addition, we assume thenumber of mobile nodes n ≤ nsnd. This notation is usedthroughout all other tables in this paper. The results aresummarized in Table 1.

Capacity Achieving Scheme I:We propose a flexible cell-partitioning scheme

to achieve a capacity that is close to the upperbound, using broadcasting and time division. Cell-partitioning schemes divide the network into severalnon-overlapping and independent cells and only allowtransmissions within the same cell. As Lemma 2 in [18]

TABLE 1The order of the optimal values of the parameters in

two-dimensional fast i.i.d. mobility model.

L: Capture Range Θ(n− 1+α+d4 / log

14 n)

R: # of Duplicates Θ(n1+α−d

2 / log12 n)

shows, each cell in the network can transmit at a rateof c3W , where c3 is a deterministic positive constant.

We group every D time-slots into a super-slot.1) At each odd super-slot, we schedule transmissions

from the sources to the relays in every time-slot.We divide the unit square into Cd = Θ

n(1−α+d)/2

logn

cells. Each cell is a square of area 1/Cd. We referto each cell in the odd super-slot as a duplicationcell. By Lemma 6 in [4], each cell can be activefor 1/c4 amount of time, where c4 is a constant.When a cell is scheduled to be active, each sourcenode in the cell broadcasts a new packet to allother nodes in the same cell for Θ

n−(2s+α−1−d)/2

log2 n

amount of time. These other nodes then serve asmobile relays for the packet. The nodes withinthe same duplication cell coordinate themselves tobroadcast sequentially.

2) At each even super-slot, we schedule transmissionsfrom the mobile relays to the destination nodesin every time-slot. We divide the unit square intoCc = Θ

n(1+α+d)/2

cells. Each cell is a square of

area 1/Cc. We refer to each cell in the even super-slot as the capture cell. In each time-slot, for eachdestination node D and each of its source nodeS, pick a node YSD that is in the same capturecell with node D in current time-slot and in thesame duplication cell with node S some time-slot inprevious super-slot and hold a copy of the packetsource node S. If there are multiple relay nodes,just pick one, which we call a representative relay,and transmit the destined packet to D. At theend of each even super-slot, clear all the buffersof mobile nodes, and prepare for a new turn ofduplication and capture.

As n → ∞, with high probability (w.h.p.), all packetsgenerated in odd duplication super-slot will finish its nd

destined transmission within the following even capturesuper-slot.

Proposition 1: With probability approaching one, asn → ∞, the above scheme allows each source to send Dpackets of size λ = Θ

n−(2s+α−1−d)/2

log2 n

to their respective

destinations within 2D time-slots.

4 TWO DIMENSIONAL I.I.D. SLOW MOBILITYMODEL

In this section, we present the upper bound on multicastcapacity-delay tradeoff under the two-dimensional i.i.d.slow mobility model, and then propose a scheme to achieve

6

a capacity close to the upper bound up to logarithmicfactors.

4.1 Upper BoundUnder slow mobility model, once a successful capturewith respect to packet p and one of its destination koccurs, the last mobile relay will start transmitting packetp to destination k within a single time slot, using possiblyother nodes as relays. Let hp,k denote the number of hopspacket p taken from the last mobile relay to destination k.And let Sh

p,k, h = 1, 2, . . . , hp,k denote the length of eachhop. Hence, similar to Lemma 2, the following lemmaholds.

Lemma 3: Under slow mobility model and concerningnetwork radio resources consumption, the following in-equality holds for any causal scheduling policy (c4 issome positive constant).

λnsTXp=1

∆2

4

E[Rp]− nd

n+

λnsTXp=1

ndXk=1

hp,kXh=1

π∆2

4E(Sh

p,k)2

≤ c5WT log n, (9)

where the sum of the hop’s lengths of the hp,i hops mustbe no smaller than the straight-line distance-captureradius:

hp,kXh=1

Shp,k ≥ Lp,k. (10)

Theorem 2: Under two-dimensional i.i.d. slow mobilitymodel, let D be the mean delay averaged over all packets,and let λ be the capacity per multicast session. Thefollowing upper bound holds for any causal schedulingpolicy,

λ = O

n log n

nsnd

3

rndD

n

. (11)

Proof: Since some of the arguments are similar toprevious sections, we only present the main proof here.

A node can either transmit or receive at one time.Therefore, it is easy to see that,

λnsTXp=1

ndXk=1

hp,kXh=1

1 ≤ WT

2n. (12)

Using Cauchy-Schwartz inequality and (12), we have

λnsTXp=1

ndXk=1

hp,kXh=1

E(Sh

p,k)2= E

λnsTXp=1

ndXk=1

hp,kXh=1

(Shp,k)

2

≥ 2

WTn

E λnsTX

p=1

ndXk=1

hp,kXh=1

Shp,k

!2

≥ 2

WTn

E λnsTX

p=1

ndXk=1

Lp,k

!2

.(13)

Equalities hold when (12) becomes an equality and Shp,k

is equal for all p, k and h.

From Lemma 1, we havendXk=1

Lp,k ≥ndXk=1

1ÈRp log nDp,k(nd − k)

=1È

Rp log n

ndXk=1

1ÈDp,k(nd − k)

≥ 1ÈRp log n

Pnd

k=1(nd − k)−14

2Pnd

k=1

pDp,k

≥ 1ÈRp log n

n32

dÈPnd

k=1 Dp,k

Pnd

k=1 1

=1È

Rp log n

ndpDp

.

(14)

Note that,λnsTXp=1

ndXk=1

Lp,k ≥λnsTXp=1

1ÈRp log n

ndpDp

≥ nd√log n

(PλnsT

p=1 Rp− 1

4 )2PλnsTp=1

pDp

≥ nd√log n

(PλnsT

p=1 Rp− 1

4 )2qPλnsTp=1 Dp

PλnsTp=1 1

= c3(PλnsT

p=1 Rp− 1

4 )2√D

,

(15)

where c3 = nd√lognλnsT

.

λnsTXp=1

E[Rp]

n+

λnsTXp=1

ndXk=1

hp,kXh=1

E(Sh

p,k)2

≥λnsTXp=1

Rp

n+

2

WTn

c23D(

λnsTXp=1

Rp− 1

4 )4.

(16)

Applying Cauchy-Schwartz inequality again and again,it can be proved that Equation (16) is greater than

Θ(q

λ3n3sT

2n2d

lognDn2 ).Hence, the optimal capacity-delay tradeoff is upper

bounded by

λ ≤ Θ

n

nsnd

3

ÊD log3 n

nnd

!.

4.2 Achievable Lower BoundChoosing Optimal Values of Key Parameters:

From Theorem 2, we have

λ = O

n

nsnd

3

ÊndD log3 n

n

= O(n− 3s+2α−2−d

3 log n).

The idea is similar, as is presented in Section 3.2. Wesummarize the optimal values in Table 2.

7

TABLE 2The order of the optimal values of the parameters in

two-dimensional slow i.i.d. mobility model.

L: Capture Range Θ(n− 1+2α+2d6 / log

12 n)

R: # of Duplicates Θ(n1+2α−d

3 )

H: # of Hops Θ(n1−α−d

3 / logn)

S: Hop Length Θ(p

logn/n)

Capacity Achieving Scheme II: We group every Dtime-slots into a super-slot. Scheme II is similar to Ca-pacity Achieving Scheme I presented in Section 3.2, andwe only introduce the differences here.

1) At each odd super-slot, we schedule transmissionsfrom the sources to the relays in every time-slot. Wedivide the unit square into Cd = Θ

n(2−2α+d)/3

logn

cell-

s. When a cell is scheduled to be active, each sourcenode in the cell broadcasts a new packet to allother nodes in the same cell for Θ

n−(3s+2α−2−d)/3

log2 n

amount of time.

2) At each even super-slot, we schedule transmissionsfrom the mobile relays to the destination nodesin every time-slot. We divide the unit square intoCc = Θ

n(1+2α+2d)/3

cells. After picking out a

representative relay, we then schedule multi-hoptransmissions in the following fashion to forwardeach packet from the representative relay to its desti-nation in the same capture cell. We further divideeach capture cell into Ch = Θ

n(2−2α−2d)/3

logn

hop-

cells (in√Ch rows and

√Ch columns). Each hop-

cell is a square of area 1/(CcCh). By Lemma 6 in[4], there exists a scheduling scheme where eachhop-cell can be active for 1/c4 amount of time.When each hop-cell is active, it forwards a packetto another node in the neighboring hop-cell. If thedestination of the packet is in the neighboring cell,the packet is forwarded directly to the destinationnode. The packets from each representative relay arefirst forwarded towards neighboring cells along theX-axis, then to their destination nodes along the Y-axis. At the end of each even super-slot, clear allthe buffers of mobile nodes, and prepare for a newturn of duplication and capture.

Proposition 2: With probability approaching one, asn → ∞, the above scheme allows each source to send Dpackets of size λ = Θ

n−(3s+2α−2−d)/3

log2 n

to their respective

destinations within 2D time-slots.

Proof: We will focus on the case in which the meandelay is bounded by a constant, i.e., D = 1. Let ⌊x⌋be the largest integer smaller than or equal to x. Weuse the following values2: Cd = ⌊(n

(2−2α)/3

8 log n )12 ⌋2, Cc =

⌊(n(1+2α)/3)12 ⌋2, Ch = ⌊(n

(2−2α)/3

4 logn )12 ⌋2. We will show that

2. To ensure the positive values, we assume α < 1.

our scheme can obtain a capacity of W32n(3s+2α−2)/3 logn

w.h.p. under multicast traffic pattern. First, we present alemma which will be used frequently in later proof. Ithas already been proven in Lemma 11, [4].

Lemma 4: Consider an experiment where we random-ly throw n balls into m ≤ n independent urns. Thesuccess probability for each ball to enter any one of theurns is p ≤ 1. Let Bi, i = 1, . . . ,m be the number of ballsin urn i after n balls are thrown. Then E[Bi] =

npm . And

as n → ∞, we have1) If np

m ≥ c log n, and c ≥ 8, then

P[Bi ≥ 2np

mfor any i] ≤ 1

n;

2) If npm ≥ cnα, where c > 0 and α > 0, then

P[Bi ≥ 2np

mfor any i] = O(

1

n);

3) If npm ≥ c log n and c ≥ 4, then

P[Bi = 0 for any i] = O(1

n).

Analysis of DuplicationWe consider the experiment in which we throw ns

balls into Cd urns with p = 1. We have

16n(3s+2α−2)/3 log n ≥ ns

Cd≥ 8n(3s+2α−2)/3 log n.

Let Nd(i) denote the number of source nodes in duplica-tion cell i. Since n ≤ nsnd, i.e., s+α ≥ 1, by Lemma 4 (1),we have

P[Nd(i) ≥ 32n(3s+2α−2)/3 log n for any i]

≤ P[Nd(i) ≥ 2ns

Cdfor any i] ≤ 1

n.

Hence, w.h.p., there are no more than32n(3s+2α−2)/3 log n source nodes within the sameduplication cell. Using time division, we can arrangeeach source to broadcast a packet for 1

32n(3s+2α−2)/3 lognamount of time in sequence.

Analysis of CaptureWe consider the experiment in which we throw n balls

into CcCh urns with p = 1. We haven

CcCh≥ 4 log n.

Let Nh(i) denote the number of nodes in hopping cell i.By Lemma 4 (3), we have

P[Nh(i) = 0 for any i] = O(1

n).

Hence, w.h.p., there is always a node in each hopping cellthat helps the multi-hop transmission.

Then we consider the experiment in which we thrown balls into CdCc urns with p = 1. We have

16 log n ≥ n

CdCc≥ 8 log n.

Let Ndc(i, j) denote the number of nodes that are induplication cell i in the previous time-slot and now in

8

capture cell j in the current time-slot. By Lemma 4 (3), wehave

P[Ndc(i, j) = 0 for any i, j] = O(1

n).

Hence, w.h.p., in each capture cell j, there is always anode which used to be in duplication cell i, and it hasall the packets broadcast in that duplication cell. If thereare multiple satisfying nodes, we only pick one for eachi as the representative relay and we get Cd representativerelays in capture cell j. On the other hand, all packetscan be found in each capture cell, and each destinationnode can find all the destined packets it desires from therepresentative relays. Hence, we only need to calculate themaximum transmissions passing through each hoppingcell if all desired transmissions are allowed.

We consider the possible transmission pairs insteadof the actual mobile nodes. A transmission pair is de-fined as the transmitting node and receiving node ina transmission. We classify the destinations based onthe sessions they belong to, i.e., one destination may becalculated multiple times when it belongs to differentsessions. Thus, there are nsnd number of pairs eitherwith different transmission nodes or requiring differentpackets.

For the transmissions horizontally passing through thehopping cell, we consider the experiment in which wethrow nsnd balls into CdCc urns with p = 1. We have

16ns+α−1 log n ≥ nsnd

CdCc≥ 8ns+α−1 log n.

Let Ns(i, j) denote the number of transmission pairswhose source nodes are located in duplication cell i anddestination nodes are located in capture cell j. By Lemma4 (1), we have

P[Ns(i, j) ≥ 32ns+α−1 log n for any i, j]

≤ P[Ns(i, j) ≥ 2nsnd

CdCcfor any i, j] ≤ 1

nsnd.

Hence, w.h.p., in each capture cell j, each representativerelay will serve no more than 32ns+α−1 log n transmissionpairs.

Since Cd representative relays are chosen in each capturecell, we consider the experiment where we throw Cd ballsinto

√Ch urns with p = 1. We have

√2n(1−α)/3

4√logn

≥ Cd√Ch

≥ n(1−α)/3

8√log n

.

Let Nr(l) denote the number of representative relays inrow l. Since α < 1, by Lemma 4 (2), we have

P[Nr(j, l) ≥√2n(1−α)/3

2√log n

for any j, l]

≤ P[Nr(j, l) ≥ 2Cd√Ch

for any j, l] = O(1

Cd) → 0.

Hence, w.h.p., the number of horizontal transmissionsTx = Ns(i, j)Nr(j, l) ≤ 16n(3s+2α−2)/3

√2 log n.

For the transmissions vertically passing through thehopping cell, we consider the experiment where we thrownsnd balls into Cc

√Ch urns with p = 1. We have

4n(3s+2α−2)/3È2 log n ≥ nsnd

Cc√Ch

≥ 2n(3s+2α−2)/3È2 log n.

Let Nch(j, l) denote the number of transmission pairswhose destinations are located in capture cell j andcolumn l. By Lemma 4 (2), we have

P[Nch(j, l) ≥ 8n(3s+2α−2)/3È2 lognfor any j, l]

≤ P[Nch(j, l) ≥ 2nsnd

Cc√Ch

for any l] = O(1

nsnd).

Hence, w.h.p., the number of vertical transmissions isTy = Nch(j, l) ≤ 8n(3s+2α−2)/3

√2 log n. And the total

transmissions passing through a single hopping cell areTx + Ty ≤ 32n(3s+2α−2)/3 log n.

5 TWO DIMENSIONAL HYBRID R.W. MOBILI-TY MODELIn this section, we study the two-dimensional hybridrandom walk mobility model with both fast and slowmobiles. We will obtain the maximum throughput forD = ω(| logB|/B2).

From Appendix G in [5], we have the following lem-ma.

Lemma 5: Under two-dimensional hybrid random walkmobility model, when given delay constraint D =ω(| logB|/B2), for any L ∈ [0, B/

√π), we have

Pr(eLp ≤ L) ≤ 36L2D, (17)

where eLp is the minimum distance between a particularmobile relay of packet p and one of its destinationswithin D time slots.

Compared with two-dimensional i.i.d. mobility model,where Pr(eLp ≤ L) = 1−(1−πL2)D ≤ πL2D, Lemma 5 isdifferent only in the coefficient, which does not influencethe orders of the final result. So following the same proofprocedure of Theorem 1 and Theorem 2, we have thefollowing two results in two-dimensional hybrid randomwalk model with fast and slow mobiles respectively.

Theorem 3: Under two-dimensional hybrid random walkfast mobility model, let D be the mean delay averagedover all packets, and let λ be the capacity per multicastsession. When B = o(1) and D = ω(| logB|/B2), thefollowing upper bound holds for any causal schedulingpolicy,

λ = O

n

nsnd

rndD

nlog3 n

. (18)

Theorem 4: Under two-dimensional hybrid random walkslow mobility model, let D be the mean delay averagedover all packets, and let λ be the capacity per multicastsession. When B = o(1) and D = ω(| logB|/B2), thefollowing upper bound holds for any causal schedulingpolicy,

λ = O

n log n

nsnd

3

rndD

n

. (19)

9

As the two-dimensional random walk mobility modelhas the same capacity upper bound as two-dimensionali.i.d. mobility model, Capacity Achieving Scheme I andScheme II still apply to R.W. mobility model with fastand slow mobiles respectively, with some extra limita-tions on delay constraint. We do not extend this partdue to space limitations.

6 ONE DIMENSIONAL I.I.D. FAST MOBILITYMODEL

6.1 Upper Bound

Lemma 6: Under one-dimensional i.i.d. mobility modeland concerning successful encounter, the following in-equality holds for any causal scheduling policy,

lognE[Dp,k] ≥1

c6E[Lp,k] +

1n

E[Rp,k]

, (20)

where c6 = 8(nd − k + 1).

Proof: To proof this lemma, we will need the follow-ing lemma on the minimum distance between the mobilerelays and the destination at any time slot. Fix a packetp that enters into the system at time slot t0(p), and oneof its destinations k. At each time slot t ≥ t0(p), let rp(t)denote the number of mobile relays holding the packetp at the beginning of the time slot t. Among these rp(t)mobile relays, there is one mobile relay whose distanceto the destination k of packet p is the smallest. Let eLp,k(t)denote this minimum distance, and let

Lp,k(t) = max 1n, eLp,k(t).

It is easy to verify that

elp,k(t) ≥ eLp,k(t) ≥ Lp,k(t)−1

n,

where elp,k(t) is the distance between a chosen relay nodeholding packet p and the destination k.

Lemma 7: Under the one-dimensional i.i.d. mobilitymodel, if n ≥ 3, then

E

"1

Lp,k(t)rp(t)|Ft−1

3

#≤ 8 log n for all t ≤ t0(p).

Proof: Let IA be the indicator function on the set A.By the definition of Lp,k(t), we have,

E

"1

Lp,k(t)|Ft−1

#= E

hnIeLp,k(t)≤ 1

n|Ft−1

i+E

"1eLp,k(t)

IeLp,k(t)>1n|Ft−1

#.

3. Let Ft be the σ-algebra that captures all information about the“history” up to time-slot t, including the nodes’ positions and thepackets they have.

Since the nodes move on a unit square, eLp,k(t) ≤√2.

Hence,

E

"1eLp,k(t)

IeLp,k(t)>1n|Ft−1

#=

1√2− nP[eLp,k(t) ≤

1

n|Ft−1]

+

Z √2

1n

1

u2P[eLp,k(t) ≤ u|Ft−1]du.

Hence,

E

"1

Lp,k(t)|Ft−1

#=

1√2+

Z √2

1n

1

u2P[eLp,k(t) ≤ u|Ft−1]du.

Under the one-dimensional i.i.d. mobility model,

P[eLp,k(t) ≤ u|Ft−1] ≤ 1− (1− 2u)(nd−k+1)rp(t)

≤ 2(nd − k + 1)urp(t).

Therefore,

E

"1

Lp,k(t)|Ft−1

#≤ c6rp(t) log n.

Finally, since rp(t) is Ft−1-measurable, we have

E

"1

Lp,k(t)rp(t)|Ft−1

#=

1

rp(t)E

"1

Lp,k(t)|Ft−1

#≤ c6 log n.

Proof of Lemma 6: Let

Vt = c6 log n[t− t0(p)]−tX

s=t0(p)+1

1

Lp,k(t)rp(t)ICp,k(t)=1,

where Cp,k(t) = 1 denotes that the scheduler decidesthat a successful capture of packet p to destination koccurs at time-slot t. Then for all t ≥ t0(p), Vt is alsoFt-measurable and Vt0(p) = 0. By Lemma 7, we have

E[Vt − Vt−1|Ft−1]

= c6 log n− E

"1

Lp,k(t, k)rp(t)ICp,k(t)=1

#

≥ c6 log n− E

"1

Lp,k(t)rp,k(t)|Ft−1

#≥ 0.

Hence,E[Vt|Ft−1] ≥ Vt−1,

i.e., Vt is a sub-martingale. Let sp,k = mint : t ≥t0(p) and Cp,k(t) = 1. Since sp,k is a stopping time, byappropriately invoking the Optional Stopping Theorem(Theorem 4.1 in [23]), we have,

E[Vsp,k ] ≥ 0.

Hence,

c6 log nE[Dp,k] ≥ E

"1

Lp,k(sp,k)Rp,k

#.

10

Using Holder’s Inequality,

E2

"1È

Lp,k(sp,k)

#≤ E[Rp,k]E

"1

Lp,k(sp,k)Rp,k

#.

Using Holder’s Inequality again, we have

c6 log nE[Dp,k] ≥1

E2[ÈLp,k(sp,k)]

1

E[Rp,k]≥ 1

E[Lp,k(sp,k)]E[Rp,k].

Finally, by definition,

lp,k = elp,k(sp,k) ≥ Lp,k(sp,k)−1

n.

Therefore,

c6 log nE[Dp,k] ≥1

(E[lp,k] + 1n )E[Rp,k]

.

Thus Lemma 6 holds,

c6 log nE[Dp] ≥1

c6E[Lp] +

1n

E[Rp]

. (21)

By Lemma 2 and Lemma 6, we have the following theo-rem.

Theorem 5: Under one-dimensional i.i.d. fast mobilitymodel, let D be the mean delay averaged over all packets,and let λ be the capacity per multicast session. WhenD = o

√n

nd

, the following upper bound holds for any

causal scheduling policy,

λ = O

n

nsnd

3

rn2dD

2

nlog5 n

. (22)

6.2 Achievable Lower BoundWe first present the optimal values of key parameters inone-dimensional i.i.d. fast mobility model in Table 3.

TABLE 3The order of the optimal values of the parameters in

one-dimensional fast i.i.d. mobility model.

L: Capture Range Θ(n− 1+α+d3 / log

13 n)

R: # of Duplicates Θ(n1+α−2d

3 / log23 n)

Capacity Achieving Scheme III:We propose a flexible rectangle-partition scheme, sim-

ilar to [5], to achieve a capacity-delay tradeoff that isclose to the upper bound. Rectangle-partition model di-vides the unit square into multiple horizontal rectangles,named as H-rectangles; and multiple vertical rectangles,named as V-rectangles as in Figure 2. A packet is said tobe destined to a rectangle if the orbit of one of its des-tinations is contained in the rectangle. Each H-rectangleand V-rectangle cross to form a cell, and transmissionsonly happen in the same crossing cell. The transmis-sion of a packet in the H(V) multicast session will go

H-r

ect

angl

e

V-rectangle

S(1)

D(1,2)

H-V Duplication

H Capture

V-H Duplication

Crossing cell

Fig. 2. One-dimensional transmissions in Scheme III.

through H(V)-V(H) duplication, V(H)-H(V) duplicationand H(V)-H(V) capture, three procedures, sequentially(see Figure 2).

We group every D time-slots into a super-slot, and letz denote any non-negative integer.

1) At each 3z + 1 super-slot, we schedule transmis-sions from the H(V)-sources to the V(H)-relays inevery time-slot. We divide the unit square intoRd H-rectangles and Rd V-rectangles, i.e., R2

d =

Θn(2−α+2d)/3

logn

crossing cells. Each cell is a square

of area 1/R2d. We refer to each cell in the 3z + 1

super-slot as a duplication cell. By Lemma 6 in [4],each cell can be active for 1/c4 amount of time,where c4 is some constant. When a cell is scheduledto be active, each H(V)-source node in the cellbroadcasts a new packet to all other V(H)-nodesin the same cell for Θ

n−(3s+α−2−2d)/3

log2 n

amount of

time. These other V(H)-nodes then serve as mobileV(H)-relays for the packet to complete the V(H)-H(V) duplications in the next super-slot. The sourcenodes within the same duplication cell coordinatethemselves to broadcast sequentially.

2) At each 3z + 2 super-slot, we schedule transmis-sions from the V(H)-relay nodes to the H(V)-relaynodes in every time-slot. We use the same partitionmethod as the one used in 3z+1 super-slot. Whena cell is scheduled to be active, search for V(H)-relay nodes holding the packet, which is destinedto the H(V)-rectangle containing this crossing celland has not been V(H)-H(V) duplicated yet. If thereare multiple satisfied V(H)-nodes for one packet,randomly choose one and broadcast the packet toall other H(V)-nodes in the same cell. We can easilyprove that with R V(H)-relay nodes for each packetp, which are generated in H(V)-V(H) duplication offormer 3z + 1 super-slot, w.h.p., there must be atime-slot within this 3z + 2 super-slot that at leastone of them reaches the destined H(V)-rectangleof packet p. And under proper scheduling, thethroughput in this period cannot be smaller thanthat in 3z + 1 super-slot.

11

3) At each 3z + 3 super-slot, we schedule transmis-sions from the mobile H(V)-relays to the H(V)-destination nodes in every time-slot. We divide theunit square into Rc = Θ

n(1+α+d)/3

H-rectangles

and Rc V-rectangles, i.e., R2c crossing cells. Each

cell is a square of area 1/R2c . We refer to each cell in

the 3z+3 super-slot as the capture cell. In each time-slot, for each H(V)-destination node D and each ofits destined packet p, search for H(V)-relay nodesin the same capture cell holding packet p. If there aremultiple ones, randomly pick one, which we call arepresentative H(V)-relay, and transmit the destinedpacket p to D. In the end of each 3z+3 super-slot,clear all the buffers of mobile nodes, and preparefor a new turn of duplications and capture.

Following the proof of Proposition 2, we haveProposition 3: With probability approaching one, as

n → ∞, the above scheme allows each source to send Dpackets of size λ = Θ

n−(3s+α−2−2d)/3

log2 n

to their respective

destinations within 3D time-slots.

7 ONE DIMENSIONAL I.I.D. SLOW MOBILITYMODEL

In this section, we study the one-dimensional i.i.d. slowmobility model.

7.1 Upper BoundBy Lemma 3 and Lemma 6, we have the following theo-rem.

Theorem 6: Under one-dimensional i.i.d. slow mobilitymodel, let D be the mean delay averaged over all packets,and let λ be the capacity per multicast session. WhenD = o

√n

nd

, the following upper bound holds for any

causal scheduling policy,

λ = O

n

nsnd

4

rn2dD

2

nlog5 n

. (23)

7.2 Achievable Lower BoundWe first present the optimal values of key parameters inone-dimensional i.i.d. slow mobility model in Table 4.

TABLE 4The order of the optimal values of the parameters in

one-dimensional slow i.i.d. mobility model.

L: Capture Range Θ(n− 1+2α+2d4 / log

34 n)

R: # of Duplicates Θ(n1+2α−2d

4 / log14 n)

H: # of Hops Θ(n1−2α−2d

4 / log54 n)

S: Hop Length Θ(p

logn/n)

Capacity Achieving Scheme IV: We group everyD time-slots into a super-slot, and let z denote anynon-negative integer. Scheme IV is similar to Capacity

Achieving Scheme III, presented in Section 6.2, and weonly introduce the differences here.

1) At each 3z + 1 super-slot, we schedule transmis-sions from the H(V)-sources to the V(H)-relays inevery time-slot. We divide the unit square intoRd H-rectangles and Rd V-rectangles, i.e., R2

d =

Θn(3−2α+2d)/4

logn

crossing cells. When a cell is sched-

uled to be active, each H(V)-source node in the cellbroadcasts a new packet to all other V(H)-nodesin the same cell for Θ

n−(4s+2α−3−2d)/4

log2 n

amount of

time.2) The same as Capacity Achieving Scheme III (2).3) At each 3z + 3 super-slot, we schedule transmis-

sions from the mobile H(V)-relays to the H(V)-destination nodes in every time-slot. We dividethe unit square into Rc = Θ

n(1+2α+2d)/4

H-

rectangles and Rc V-rectangles, i.e., R2c crossing

cells. After picking out a representative H(V)-relay,we then schedule multi-hop transmissions in thefollowing fashion to forward this destined packetp from the representative H(V)-relay to D. We furtherdivide each capture cell into Rh = Θ

n(1−2α−2d)/4√

logn

H-rectangles and Rh V-rectangles, i.e., R2

h crossinghop-cells. Each hop-cell is a square of side length1/(RcRh). By Lemma 6 in [4], there exists a schedul-ing scheme where each hop-cell can be active for1/c4 amount of time. When each hop-cell is active,it forwards a packet to another H(V)-node in theneighboring hop-cell. If the H(V)-destination nodeof the packet is in the neighboring cell, the packetis forwarded directly to the H(V)-destination node.The packets from each representative H(V)-relay arefirst forwarded towards neighboring cells along theX-axis, then to their destination nodes along the Y-axis. At the end of each 3z + 3 super-slot, clear allthe buffers of mobile nodes, and prepare for a newturn of duplications and capture.

Proposition 4: With probability approaching one, asn → ∞, the above scheme allows each source to send Dpackets of size λ = Θ

n−(4s+2α−3−2d)/4

log2 n

to their respective

destinations within 3D time-slots.

8 ONE DIMENSIONAL HYBRID R.W. MOBILI-TY MODEL

In this section, we present the optimal multicastcapacity-delay tradeoffs of the one-dimensional hybridrandom walk mobility model with both fast and slowmobiles. The results can be established by following sim-ilar analysis as one-dimensional i.i.d. mobility models.The details are omitted here for brevity.

Theorem 7: Under one-dimensional hybrid random walkfast mobility model, let D be the mean delay averagedover all packets, and let λ be the capacity per multicastsession. When B = o(1) and D = ω(1/B2), the following

12

upper bound holds for any causal scheduling policy,

λ = O

n

nsnd

3

rn2dD

2

nlog5 n

. (24)

Theorem 8: Under one-dimensional hybrid random walkslow mobility model, let D be the mean delay averagedover all packets, and let λ be the capacity per multicastsession. When B = o(1) and D = ω(1/B2), the followingupper bound holds for any causal scheduling policy,

λ = O

n

nsnd

4

rn2dD

2

nlog5 n

. (25)

9 RESULTS DISCUSSIONS

Our results of optimal multicast capacity-delay tradeoffsin mobile ad hoc networks give a more general resultthan previous works:

• It generalizes the optimal delay-throughput trade-offs in unicast traffic pattern in [5], when we setns = n and nd = 1.

• It generalizes the multicast capacity resultO(ÈD/ns) under delay constraint in [15], which

considers the two-dimensional i.i.d. fast mobilitymodel and provides better results than [14], whenwe set nsnd = n.

We summarize our results in Table 5 where we omitthe logarithmic factors. Setting ns = n and nd = 1, ourresults are shown in the second column. Setting ns = nand nd = k, our results are shown in the third column.

TABLE 5Optimal multicast capacity and delay tradeoffs in

MANETs: a global perspective

λ (i.i.d./hybrid r.w.) unicast multicast

2D fast mobility O

ÈDn

O

1k

ÈDnk

2D slow mobility O

3

ÈDn

O

1k

3

ÈDnk

1D fast mobility O

3

ÈD2

n

O

1k

3

ÈD2

nk2

1D slow mobility O

4

ÈD2

n

O

1k

4

ÈD2

nk2

We would like to mention that, similar to the unicastcase, [5], our one-dimensional mobility models achievea larger capacity than two-dimensional models underthe multicast traffic pattern. The advantage of lowerdimensional mobility lies in the fact that it is simpleand easily predictable, thus increasing the inter con-tact rate. Though nodes are limited to only movinghorizontally or vertically, the mobility range on theirorbit lines is not restricted. Moreover, for H(V) multicastsessions, the V(H)-relay nodes are used to compensatefor the lack of vertical(horizontal) mobility. Given theabove analysis, the one-dimensional mobility model inour paper is actually a hybrid dimensional model, where

one-dimensional mobile nodes transmit packets in two-dimensional space. We plan to study the capacity im-provement brought about by this hybrid dimensionalmodel in the future.

10 HETEROGENEOUS NETWORKS WITH IN-FRASTRUCTURE SUPPORT

In heterogeneous networks, transmissions can be carriedout either in infrastructure mode or in ad hoc mode (seeFigure 3). Let Ti and Ta denote the maximum aggregateinput throughput of the whole network when all thetransmissions are carried out in infrastructure mode andin ad hoc mode, respectively. Then, the aggregate inputcapacity of heterogeneous wireless networks, denoted byT , can be calculated as follows:

T = maxTi, Ta. (26)

Infrastructure Mode (Ti)

Ad hoc Mode (Ta)

Uplink

(Ti ≤ mWi/2) Downlink

(qTi ≤ mWi/2)T

BS

S

R D

Aggregate

Throughput

Fig. 3. Two modes in heterogeneous networks.

Ta has been studied in previous sections under dif-ferent mobility models. We only discuss Ti here. Sinceall base stations share information simultaneously, theycan be regarded as an integrated relay. We define thisspecific huge relay as BS relay, with the same maximuminput and output throughput mWi/2. Please recall thatm = nβ is the number of base stations in the network.

Under infrastructure mode, packets should be trans-mitted by three steps. Packets is sent from source tothe nearest base station first and then shared by all thebase stations. At the third step, packets are sent to thedestination by the nearest base station. Since we assumethe base stations are wired connected, the bandwidth canbe regarded as large enough and the delay at the secondstep is ignored.

To conduct further analysis, we present a lemma firstto bound the number of nodes in a given region.

Lemma 8: For each packet and in each subregion,w.h.p., there are at most Θ(nd

m ) destination nodes whenm = o(nd), and at most Θ(1) destination nodes whenm = ω(nd).

Proof: Consider subregion i. Let Xi be a randomvariable that denotes the number of destination nodes

13

in subregion i. Then we have E[Xi] = nd

m . Recall theChernoff Bound in [24], for any δ > 0,

PrXi > (1 + δ)E[Xi]

< e−E[Xi]f(δ) (27)

where f(δ) = (1 + δ) log(1 + δ)− δ.1) m = o(nd), i.e., 0 ≤ β < α ≤ 1.

According to the Chernoff bound (27), we have

Pr(Xi > 2nd

m) < e−

ndm f(1)

where f(1) = 2 log 2−1 > 0. Since 0 ≤ β < α, whenn is large enough, we have

Pr(Xi ≤ 2nd

mfor any i ) ≥ 1−mPr(Xi > 2

nd

m)

> 1− nβe−nα−βf(1)

→ 1

2) m = ω(nd), i.e., 0 ≤ α < β ≤ 1.According to the Chernoff bound (27), we have

Pr(Xi > (1 + δ)nd

m) <

eδndm

(1 + δ)(1+δ)ndm

.

Let (1 + δ)nd

m = c7, where c7 is a constant that willbe determined later. Then we have

Pr(Xi > c7) <e2−nα−β

c7nc7(β−α).

Hence,

Pr(Xi ≤ c7 for any i ) > 1− e2−nα−β

c7n(c7−1)β−c7α.

We can choose c7 > ββ−α , so that (c7−1)β−c7α > 0.

This probability goes to 1 as n goes to infinity.

For the first step, when m = Θ(ns), there will be atmost log n source nodes inside every subregion. There-fore, the aggregate input throughput for the first step isTi1 = Θ( m

logn ). When m = ω(ns), there will be at mostΘ(1) source nodes inside every subregion and Ti1 =Θ(ns). When m = o(ns), there will be at least Θ(ns

m )source nodes inside every subregion and Ti1 = Θ(m).

Under multicast traffic pattern, multiple deliveries areneeded to reach all the nd destinations. Hence, for eachinput flow of the BS relay, there are at least q outputflows. Note that q = nd, for there may be multipledestination nodes within the same subregion and onlyone broadcast is enough for all of them.

For the third step, if m = ω(nd), there will be atmost constant destinations in every subregion for eachmulticast session. Since nsnd ≥ n ≥ m, there will be atleast Θ(nsnd

m ) output flows for each base station. Hence,the aggregate input throughput for the third step isTi2 = Θ( m2

nsnd). When m = o(nd), there are at least

Θ(nd

m ) destinations for every multicast session in everysubregion, which means that the base station shouldbroadcast the information for every multicast sessionand therefore Ti2 = Θ(m).

Combined the above results together, we can obtainthe aggregate input throughput for the network. It canalso be inferred that base stations may not help unlessthere are a large number of them.

11 CONCLUSION

In this paper, we have studied the multicast capacity-delay tradeoffs in both homogeneous and hetero-geneous mobile networks. Specifically, in homoge-neous networks, we established the upper bound onthe optimal multicast capacity-delay tradeoffs undertwo-dimensional/one-dimensional i.i.d./hybrid randomwalk fast/slow mobility models and proposed capaci-ty achieving schemes to achieve capacity close to theupper bound. In addition, we find that though the one-dimensional mobility model constrains the direction ofnodes’ mobility, it achieves larger capacity than the two-dimensional model since it is more predictable. Also,slow mobility brings better performance than fast mo-bility because there are more possible routing schemes.

12 ACKNOWLEDGMENT

This paper is supported by National Fundamen-tal Research Grant (No. 2011CB302701); NSF China(No. 61271219,61202373); Shanghai Basic Research KeyProject (No. 11JC1405100); China Ministry of Educa-tion Fok Ying Tung Fund (No. 122002); China Min-istry of Education New Century Excellent Talent (No.NCET-10-0580); China Postdoctoral Science FoundationGrant 2011M500774 and 2012T50417, STCSM Grant12JC1405200 and SEU SKL project #2012D13.

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Jinbei Zhang received his B. E. degree in Elec-tronic Engineering from Xidian University, Xi’an,China, in 2010, and is currently pursuing thePh.D. degree in electronic engineering at Shang-hai Jiao Tong University, Shanghai, China.

His current research interests include networksecurity, capacity scaling law and mobility mod-els in wireless networks.

Xinbing Wang received the B.S. degree (withhons.) from the Department of Automation,Shanghai Jiaotong University, Shanghai, China,in 1998, and the M.S. degree from the De-partment of Computer Science and Technology,Tsinghua University, Beijing, China, in 2001. Hereceived the Ph.D. degree, major in the Depart-ment of electrical and Computer Engineering,minor in the Department of Mathematics, NorthCarolina State University, Raleigh, in 2006. Cur-rently, he is a faculty member in the Department

of Electronic Engineering, Shanghai Jiaotong University, Shanghai,China. His research interests include scaling law of wireless networksand cognitive radio. Dr. Wang has been an associate editor for IEEETransactions on Mobile Computing, and the member of the TechnicalProgram Committees of several conferences including ACM MobiCom2012, ACM MobiHoc 2012, IEEE INFOCOM 2009-2013.

Xiaohua Tian received his B.E. and M.E. de-grees in communication engineering from North-western Polytechnical University, Xi’an, China,in 2003 and 2006, respectively. He received thePh.D. degree in the Department of Electrical andComputer Engineering (ECE), Illinois Institute ofTechnology (IIT), Chicago, in Dec. 2010. He iscurrently a research associate in Departmen-t of Electronic Engineering in Shanghai Jiao-tong University, China. His research interestsinclude application-oriented networking, Internet

of Things and wireless networks. He serves as the guest editor ofInternational Journal of Sensor Networks, publicity co-chair of the7th International Conference on Wireless Algorithms, Systems, andApplications (WASA 2012). He also serves as the Technical ProgramCommittee member for Communications QoS, Reliability, and ModelingSymposium (CQRM) of GLOBECOM 2011, Wireless Networking ofGLOBECOM 2013 and WASA 2011.

PLACEPHOTOHERE

Y un Wang received the BE degree in electronicengineering in Shanghai Jiao Tong University,China in 2011. His research interests are in thearea of asymptotic analysis of multicast capacityin wireless ad hoc network.

PLACEPHOTOHERE

X iaoyu Chu received the BE degree in electronicengineering in Shanghai Jiao Tong University,China in 2010. Her research interests are in thearea of asymptotic analysis of multicast capacityin wireless ad hoc network.

Yu Cheng received the B.E. and M.E. degrees inElectrical Engineering from Tsinghua University,Beijing, China, in 1995 and 1998, respectively,and the Ph.D. degree in Electrical and Com-puter Engineering from the University of Water-loo, Waterloo, Ontario, Canada, in 2003. FromSeptember 2004 to July 2006, he was a post-doctoral research fellow in the Department ofElectrical and Computer Engineering, Universityof Toronto, Ontario, Canada. Since August 2006,he has been with the Department of Electrical

and Computer Engineering, Illinois Institute of Technology, Chicago,Illinois, USA, as an Assistant Professor. His research interests includenext-generation Internet architectures and management, wireless net-work performance analysis, network security, and wireless/wireline inter-working. He received a Postdoctoral Fellowship Award from the NaturalSciences and Engineering Research Council of Canada (NSERC) in2004, and a Best Paper Award from the conferences QShine 2007and ICC 2011. He received the National Science Foundation (NSF)CAREER award in 2011. He served as a Co-Chair for the WirelessNetworking Symposium of IEEE ICC 2009, a Co-Chair for the Communi-cations QoS, Reliability, and Modeling Symposium of IEEE GLOBECOM2011, and a Technical Program Committee (TPC) Co-Chair for WASA2011. He is an Associated Editor for IEEE Transactions on VehicularTechnology.